 
C
C                     * * * * * * * * * * * * * *
C                     *                         *
C                     *         VSFFTPK         *
C                     *                         *
C                     * * * * * * * * * * * * * *
C
C
C                      (VERSION 1,  AUGUST 1985)
C
C       A VECTORIZED PACKAGE OF FORTRAN SUBPROGRAMS FOR THE FAST
C   TRANSFORM OF MULTIPLE REAL SEQUENCES DEFINED ON A STAGGERED GRID
C
C                                  BY
C
C                LINDA L. LINDGREN AND ROLAND A. SWEET
C
C                    SCIENTIFIC COMPUTING DIVISION
C                     NATIONAL BUREAU OF STANDARDS
C                       BOULDER, COLORADO 80303
C
C
C                     * * * * * * * * * * * * * *
C                     *                         *
C                     *        PURPOSE          *
C                     *                         *
C                     * * * * * * * * * * * * * *
C
C
C     THIS PACKAGE HAS BEEN WRITTEN TO PERFORM FAST TRANSFORMS OF
C     DISCRETE FUNCTIONS DEFINED ON A 'STAGGERED' GRID, I.E. A GRID ON
C     WHICH BOUNDARY VALUES ARE IMPOSED, NOT AT A GRID POINT, BUT AT A
C     POINT MIDWAY BETWEEN TWO GRID POINTS.  SUCH TRANSFORMS MAY BE
C     USED TO EFFICIENTLY SOLVE POISSON-TYPE PARTIAL DIFFERENTIAL
C     EQUATIONS.  EACH TRANSFORM IS DESCRIBED IN THE DOCUMENTATION FOR
C     EACH INDIVIDUAL ROUTINE.
C
C
C         NOTATION:  TO SIMULATE THE MATHEMATICAL SYMBOL 'SIGMA'
C         THAT REPRESENTS SUMMATION, WE DEFINE THE NOTATION
C
C         SUM(L=IS,IF)[ Y(L) ] = Y(IS) + Y(IS+1) + . . . + Y(IF) .
C
C
C
C              * * * * * * * * * * * * * * * * * * * * *
C              *                                       *
C              *         DESCRIPTION OF PACKAGE        *
C              *                                       *
C              * * * * * * * * * * * * * * * * * * * * *
C
C
C     THIS PACKAGE CONSISTS OF SEVERAL SUBROUTINES THAT PERFORM FAST
C     STAGGERED GRID TRANSFORMS ON MULTIPLE REAL SEQUENCES.  EACH
C     TRANSFORM ROUTINE CONSISTS OF THREE TASKS: A PRE-PROCESSING STEP
C     THAT COMPUTES A NEW SEQUENCE FROM THE GIVEN DATA, CALLS THE
C     APPROPRIATE ROUTINE IN THE VECTORIZED FAST FOURIER TRANSFORM
C     PACKAGE VRFFTPK TO CALCULATE AN FFT OF THE NEW SEQUENCE, AND THEN
C     AS A POST-PROCESSING STEP COMPUTES THE REQUIRED TRANSFORM FROM THE
C     REAL PERIODIC TRANSFORM.  EACH TRANSFORM ROUTINE HAS AN ASSOCIATED
C     INITIALIZATION ROUTINE THAT MUST BE CALLED BEFORE THE TRANSFORM.
C
C     THE USER-CALLABLE SUBROUTINES ARE:
C
C     1.  VSRFTI     INITIALIZATION ROUTINE FOR VSRFTF AND VSRFTB
C     2.  VSCOSI     INITIALIZATION ROUTINE FOR VSCOSF AND VSCOSB
C     3.  VSSINI     INITIALIZATION ROUTINE FOR VSSINF AND VSSINB
C                    (THIS ROUTINE EXISTS ONLY AS AN ENTRY POINT IN
C                    VSCOSI.)
C     4.  VSCSQI     INITIALIZATION ROUTINE FOR VSCOSQ
C     5.  VSSNQI     INITIALIZATION ROUTINE FOR VSSINQ (THIS ROUTINE
C                    EXISTS ONLY AS AN ENTRY POINT IN VSCSQI.)
C     6.  VSRFTF     ANALYSIS (FORWARD TRANSFORM) OF MULTIPLE REAL
C                    STAGGERED GRID SEQUENCES INTO FOURIER COEFFICIENTS
C     7.  VSRFTB     SYNTHESIS (BACKWARD TRANSFORM) OF MULTIPLE
C                    FOURIER COEFFICIENTS INTO REAL STAGGERED GRID
C                    SEQUENCES
C     8.  VSCOSF     ANALYSIS (FORWARD TRANSFORM) OF MULTIPLE REAL
C                    STAGGERED GRID SEQUENCES INTO FOURIER COSINE
C                    COEFFICIENTS
C     9.  VSCOSB     SYNTHESIS (BACKWARD TRANSFORM) OF MULTIPLE FOURIER
C                    COSINE COEFFICIENTS INTO REAL STAGGERED GRID
C                    SEQUENCES
C    10.  VSCOSQ     ANALYSIS AND SYNTHESIS (FORWARD AND BACKWARD
C                    TRANSFORM) OF MULTIPLE REAL STAGGERED GRID
C                    SEQUENCES INTO FOURIER COSINE QUARTER-WAVE
C                    COEFFICIENTS
C    11.  VSSINF     ANALYSIS (FORWARD TRANSFORM) OF MULTIPLE REAL
C                    STAGGERED GRID SEQUENCES INTO FOURIER SINE
C                    COEFFICIENTS
C    12.  VSSINB     SYNTHESIS (BACKWARD TRANSFORM) OF MULTIPLE FOURIER
C                    SINE COEFFICIENTS INTO REAL STAGGERED GRID
C                    SEQUENCES
C    13.  VSSINQ     ANALYSIS AND SYNTHESIS (FORWARD AND BACKWARD
C                    TRANSFORM) OF MULTIPLE REAL STAGGERED GRID
C                    SEQUENCES INTO FOURIER SINE QUARTER-WAVE
C                    COEFFICIENTS
C
C     ALL VECTOR LENGTHS ARE EQUAL TO THE NUMBER OF VECTORS BEING
C     TRANSFORMED.
C
 
 
